Theorem mobii 2114

Description: Formula-building rule for "at most one" quantifier (inference form). (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 17-Oct-2016.)

Hypothesis

Ref	Expression
mobii.1	⊢ (𝜓 ↔ 𝜒)

Assertion

Ref	Expression
mobii	⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

Proof of Theorem mobii

Step	Hyp	Ref	Expression
1		mobii.1	. . . 4 ⊢ (𝜓 ↔ 𝜒)
2	1	a1i 9	. . 3 ⊢ (⊤ → (𝜓 ↔ 𝜒))
3	2	mobidv 2113	. 2 ⊢ (⊤ → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒))
4	3	mptru 1404	1 ⊢ (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)

Syntax hints:   ↔ wb 105  ⊤wtru 1396  ∃*wmo 2078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-eu 2080  df-mo 2081

This theorem is referenced by:  moaneu  2154  moanmo  2155  2moswapdc  2168  2exeu  2170  rmobiia  2722  rmov  2821  euxfr2dc  2989  rmoan  3004  2rmorex  3010  mosn  3703  dffun9  5353  funopab  5359  funco  5364  funcnv2  5387  funcnv  5388  fun2cnv  5391  fncnv  5393  imadif  5407  fnres  5446  ovi3  6154  oprabex3  6286  axaddf  8078  axmulf  8079  frecuzrdgtcl  10664  frecuzrdgfunlem  10671  fsum3  11938